[Abridged] Combining HST/WFPC2 mosaics with extensive ground-based spectroscopy, we study the nature of E+A galaxies in three intermediate redshift clusters (z=0.33, 0.58, & 0.83). From a sample of ~500 confirmed cluster members, we isolate 46 E+A candidates to determine the E+A fraction and study their physical properties. We find E+A's comprise a non-negligible component (~7-13%) of the cluster population at these redshifts, and their diverse nature indicates a heterogeneous parent population. Cluster E+A's span the range in Hubble type and cover a wide range in luminosity, internal velocity dispersion, and half-light radius. From their velocity dispersions and half-light radii, we infer that the descendants of E+A's in our highest redshift cluster are massive early-type galaxies. We find a decrease in the characteristic E+A mass similar to the decrease in luminosity of rapidly star-forming field galaxies since z~1, i.e. galaxy ``down-sizing.'' In addition, we argue our statistics imply that >30% of the E-S0 members have undergone an E+A phase; the true fraction could be 100% if the effects of E+A down-sizing, an increasing E+A fraction with redshift, and the conversion of spirals into early-types are also considered. Thus, the E+A phase may indeed be an important stage in the transformation of star-forming galaxies into early-type cluster members.

Accepted for publication in ApJ, version with higher resolution images available at http://www.exp-astro.phys.ethz.ch/tran/publications.html